4
$\begingroup$

How accurate is the following statement: "For tree graphs, the multiplicity of the smallest non-zero eigenvalue $\lambda_2$ of the Laplacian is 1." If not valid, in which cases does it fail to hold?

I am particularly interested in understanding the connection between tree structures and the multiplicity of $\lambda_2$.

Improve this question
$\endgroup$

2 Answers 2

Reset to default
4
$\begingroup$

The multiplicity of Laplacian eigenvalues of tree graphs is studied in arXiv:1907.11482. If $\Delta$ is the maximal degree of the graph, and $\Delta\geq 2$, then the multiplicity $m_2$ of $\lambda_2$ is bounded by $m_2\leq \Delta-1$ (proposition 2.8).

Improve this answer
$\endgroup$
4
$\begingroup$

All the Laplacian eigenvalues of the star graph $K_{1,n}$ other than $n+1$ and $0$ are equal to $1$.

Computer experimentation reveals a modest number of additional examples. One pattern that may generalize (although I have not bothered to do so) is to form a graph consisting of $m$ paths of length $k$ glued together at a single vertex.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .